about the coordinate system of covariance and conic matrix

[djx99]

I am currently trying to learn and understand your code. I noticed that the inverse covariance matrix computed by the project_gaussians_2d_scale_rot function is inconsistent with the result derived from the traditional formula. It seems like this might be related to a difference in the coordinate system (e.g., between a left-handed and right-handed system).

I also looked into the source code of the project_gaussians_2d_scale_rot_forward_kernel, rotmat2d, and scale_to_mat2d functions. From what I can see, your calculations seem consistent with the traditional formula. However, the computed result is still different. Specifically, your result is [0.1, 0.06, 0.1], while mine is [0.1, -0.06, 0.1].

Could you please clarify which coordinate system your implementation uses? And why do the results differ despite the calculations appearing to follow the traditional formula?

import torch
from gsplat.project_gaussians_2d_scale_rot import project_gaussians_2d_scale_rot
if name == "main":
  device = torch.device("cuda")
  xy = torch.tensor([[0, 0]]).float().to(device)
  scale = torch.tensor([[5, 2.5]]).float().to(device)
  rotation = torch.tensor([[-0.7854]]).float().to(device)
  H, W, BLOCK_H, BLOCK_W = 100, 100, 16, 16
  tile_bounds = (
  (W + BLOCK_W - 1) // BLOCK_W,
  (H + BLOCK_H - 1) // BLOCK_H,
  1,
  ) 
  _, _, _, conics, _ = project_gaussians_2d_scale_rot(xy, scale, rotation, H, W, tile_bounds)
  print(conics)
  
  R = torch.zeros(size=(1, 2, 2), device=device)
  S = torch.zeros(size=(1, 2, 2), device=device)
  R[:, 0, 0] = torch.cos(rotation[:, 0])
  R[:, 0, 1] = -torch.sin(rotation[:, 0])
  R[:, 1, 0] = torch.sin(rotation[:, 0])
  R[:, 1, 1] = torch.cos(rotation[:, 0])
  S[:, 0, 0] = scale[:, 0]
  S[:, 1, 1] = scale[:, 1]
  RS = R @ S
  RST = torch.transpose(RS, 1, 2)
  cov = RS @ RST
  # conic = torch.linalg.inv(cov)
  # print(conic)
  conic = torch.zeros(size=(1, 2, 2), device=device)
  det = cov[:, 0, 0] * cov[:, 1, 1] - cov[:, 0, 1] ** 2
  inv_det = 1 / det
  conic[:, 0, 0] = cov[:, 1, 1] * inv_det
  conic[:, 0, 1] = -cov[:, 0, 1] * inv_det
  conic[:, 1, 0] = -cov[:, 0, 1] * inv_det
  conic[:, 1, 1] = cov[:, 0, 0] * inv_det
  print(conic)